Related objects

Learn more about

Show commands for: Magma / SageMath

Decomposition of \( S_{12}^{\mathrm{new}}(15) \) into irreducible Hecke orbits

magma: S := CuspForms(15,12);
magma: N := Newforms(S);
sage: N = Newforms(15,12,names="a")
Label Dimension Field $q$-expansion of eigenform
15.12.1.a 1 \(\Q\) \(q \) \(\mathstrut-\) \(56q^{2} \) \(\mathstrut-\) \(243q^{3} \) \(\mathstrut+\) \(1088q^{4} \) \(\mathstrut+\) \(3125q^{5} \) \(\mathstrut+\) \(13608q^{6} \) \(\mathstrut+\) \(27984q^{7} \) \(\mathstrut+\) \(53760q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)
15.12.1.b 2 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(243q^{3} \) \(\mathstrut+\) \(\bigl(- 22 \alpha_{2} \) \(\mathstrut- 560\bigr)q^{4} \) \(\mathstrut-\) \(3125q^{5} \) \(\mathstrut+\) \(243 \alpha_{2} q^{6} \) \(\mathstrut+\) \(\bigl(- 1276 \alpha_{2} \) \(\mathstrut- 19468\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 2124 \alpha_{2} \) \(\mathstrut- 32736\bigr)q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)
15.12.1.c 2 $\Q(\alpha_{ 3 })$ \(q \) \(\mathstrut+\) \(\alpha_{3} q^{2} \) \(\mathstrut-\) \(243q^{3} \) \(\mathstrut+\) \(\bigl(- 13 \alpha_{3} \) \(\mathstrut- 1640\bigr)q^{4} \) \(\mathstrut-\) \(3125q^{5} \) \(\mathstrut-\) \(243 \alpha_{3} q^{6} \) \(\mathstrut+\) \(\bigl(3584 \alpha_{3} \) \(\mathstrut+ 27188\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- 3519 \alpha_{3} \) \(\mathstrut- 5304\bigr)q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)
15.12.1.d 3 $\Q(\alpha_{ 4 })$ \(q \) \(\mathstrut+\) \(\alpha_{4} q^{2} \) \(\mathstrut+\) \(243q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{4} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(3125q^{5} \) \(\mathstrut+\) \(243 \alpha_{4} q^{6} \) \(\mathstrut+\) \(\bigl(- 12 \alpha_{4} ^{2} \) \(\mathstrut+ 460 \alpha_{4} \) \(\mathstrut+ 38888\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(- \alpha_{4} ^{2} \) \(\mathstrut+ 1354 \alpha_{4} \) \(\mathstrut- 7248\bigr)q^{8} \) \(\mathstrut+\) \(59049q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 2 })\cong$ \(\Q(\sqrt{1609}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 22 x \) \(\mathstrut -\mathstrut 1488\)
$\Q(\alpha_{ 3 })\cong$ \(\Q(\sqrt{1801}) \) \(x ^{2} \) \(\mathstrut +\mathstrut 13 x \) \(\mathstrut -\mathstrut 408\)
$\Q(\alpha_{ 4 })$ \(x ^{3} \) \(\mathstrut +\mathstrut x ^{2} \) \(\mathstrut -\mathstrut 5450 x \) \(\mathstrut +\mathstrut 7248\)

Decomposition of \( S_{12}^{\mathrm{old}}(15) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(15) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/5/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(5)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/3/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(3)) }^{\oplus 2 }\oplus \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 4 } $